Should I use minimax or alpha-beta pruning?
Both algorithms should give the same answer. However, their main difference is that alpha-beta does not explore all paths, like minimax does, but prunes those that are guaranteed not to be an optimal state for the current player, that is max or min. So, alpha-beta is a better implementation of minimax.

Here are the time complexities of both algorithms


Minimax: $\mathcal{O}(b^d)$,  
Alpha-beta (best-case scenario): $\mathcal{O}(b^{d/2})  = \mathcal{O}(\sqrt{b^d})$ 


where $b$ is an average branching factor with a search depth of $d$ plies.